graph theory A branch of mathematics used to represent relations and networks. A graph consists of a set of points (nodes or vertices) and the pairwise links between them (arcs or lines). In sociological applications, the nodes are typically individuals, roles, or organizations, and the links are social relationships (such as kinship, friendship, communication, or authority). The links may take account of direction (a digraph) or ignore it (an undirected graph) and they may be at any level of measurement. Apart from the usual binary graph (where a link either exists or does not), special cases of interest include asymmetric graphs for representing tournaments, ordered graphs and lattices for organizational structures, trees for hierarchies and classification systems, signed (+, −) graphs for structural balance, real-valued graphs for distribution and assignment problems, and links which express the probability of a relationship (stochastic graphs).

Graph theory provides theorems (proved consequences) and algorithms (step-by-step procedures) used to obtain information, such as properties of individual points (popularity, centrality, liaison, or bridge status), of pairs (shortest path between two points) or subgroups (clique-detection, triads), and sub-graphs (blocks of points which are ‘structurally equivalent’ by having the same pattern of links).

Earliest sociological uses include sociometry and clique-detection, and more recently social network, kinship and citation structures, and ‘vacancy chains’ tracing the movement of occupational or housing vacancies through a system. Graph theory has also been extended to treat very large networks and to compare actual network structure with randomly constructed graphs.